R\'enyi's $\alpha$-divergence variational Bayes for spike-and-slab high-dimensional linear regression

Abstract

Sparse high-dimensional linear regression is a central problem in statistics, where the goal is often variable selection and/or coefficient estimation. We propose a mean-field variational Bayes approximation for sparse regression with spike-and-slab Laplace priors that replaces the standard Kullback-Leibler (KL) divergence objective with the R\'enyi's $\alpha$ divergence: a one-parameter generalization of KL divergence indexed by $\alpha \in (0, \infty) \setminus \{1\}$ that allows flexibility between zero-forcing and mass-covering behavior. We derive coordinate ascent variational inference (CAVI) updates via a second-order delta method and develop a stochastic variational inference algorithm based on a Monte Carlo surrogate R\'enyi lower bound. In simulations, our two methods perform comparably to state-of-the-art Bayesian variable selection procedures across a range of sparsity configurations and $\alpha$ values for both variable selection and estimation, and our numerical results illustrate how different choices of $\alpha$ can be advantageous in different sparsity configurations.